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Mirrors > Home > MPE Home > Th. List > Mathboxes > psubssat | Structured version Visualization version GIF version |
Description: A projective subspace consists of atoms. (Contributed by NM, 4-Nov-2011.) |
Ref | Expression |
---|---|
atpsub.a | ⊢ 𝐴 = (Atoms‘𝐾) |
atpsub.s | ⊢ 𝑆 = (PSubSp‘𝐾) |
Ref | Expression |
---|---|
psubssat | ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ∈ 𝑆) → 𝑋 ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
2 | eqid 2821 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
3 | atpsub.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | atpsub.s | . . 3 ⊢ 𝑆 = (PSubSp‘𝐾) | |
5 | 1, 2, 3, 4 | ispsubsp 36896 | . 2 ⊢ (𝐾 ∈ 𝐵 → (𝑋 ∈ 𝑆 ↔ (𝑋 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑋 ∀𝑞 ∈ 𝑋 ∀𝑟 ∈ 𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ 𝑋)))) |
6 | 5 | simprbda 501 | 1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ∈ 𝑆) → 𝑋 ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ⊆ wss 3936 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 lecple 16572 joincjn 17554 Atomscatm 36414 PSubSpcpsubsp 36647 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-ov 7159 df-psubsp 36654 |
This theorem is referenced by: psubatN 36906 paddidm 36992 paddclN 36993 paddss 36996 pmodlem1 36997 pmod1i 36999 pmodl42N 37002 elpcliN 37044 pclidN 37047 pclbtwnN 37048 pclunN 37049 pclun2N 37050 pclfinN 37051 polssatN 37059 psubclsubN 37091 |
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